They played their role with gusto: hissing, guffawing or cackling when Mrs. Clinton spoke; whooping and air-punching for Mr. Trump. Ronald Reagan, the patron saint of modern Republicanism, grinned from a framed portrait in the corner. Pizza was served.

It seemed as good of a place as any to ask a question that has baffled so many observers outside the United States: Why do so many Americans support Mr. Trump?

Readers emailing from abroad are surprised by the campaign’s bluster, insults and prevarications. They don’t understand how a race for the American presidency can be dominated by arguments over missing emails, “birtherism” and beauty queens.

Across the United States, though, the campaign is about much more than that. I’ve spent the past 10 days talking to Trump supporters, looking for answers.

Arizona, with its searing deserts and saw-toothed mountains, has long been a Republican bastion, a place of conservative-leaning politics and liberal gun laws. The state’s most prominent Trump advocate is Sheriff Joe Arpaio of Maricopa County, who became notorious for his intolerance of unauthorized immigrants and making prison inmates wear pink underwear.

At the debate party in Phoenix, the largely white crowd was seized by a palpable air of anticipation. After a sharp dip in August, Mr. Trump was suddenly riding high again in the polls. Could he pull ahead of Mrs. Clinton?

The first thunderclap of applause erupted when Mr. Trump said that Mrs. Clinton has been “doing this for 30 years,” with little to show for her efforts. But as the debate wore on, and Mr. Trump’s energy started to flag, so too did the atmosphere in the room. Onscreen, Mrs. Clinton grew in confidence, unleashing a volley of attack lines. As Mr. Trump struggled to respond, the crowd in Arizona lapsed into ever longer silences.

A supporter of Mr. Trump in the crowd who refused to be identified, citing unspecified threats from Clinton loyalists, said he didn’t expect his candidate to be as polished as a professional politician. He was willing to take the brash Mr. Trump as is, he said.

Others at the viewing party said they were worried by Mr. Trump’s performance. Kathy Manie, wearing a red “Make America Great Again” hat, said she was disappointed that Mr. Trump appeared ill-prepared for Mrs. Clinton’s rehearsed barbs. “He could have spoken more eloquently,” she said. “Hillary knew exactly how to taunt him, and he fell for the bait. He gave as much as he got, but I wish his words were stronger.”

Ms. Manie was not the prototypical Trump supporter. Born in Trinidad, she immigrated to the United States more than two decades ago, at 21. She is a social worker and a committed Christian, and she said that she was praying for Mr. Trump to win.

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Supporters of Mr. Trump at a rally on Tuesday in Melbourne, Fla. A common sentiment among them: Their man fuses a potent anti-establishment vim with a certain brand of American authenticity.Credit
Damon Winter/The New York Times

“I can’t stand another four years of a Democrat running this country,” she said, but she worried the forces of history were pulling against her preference. “We’re at a time when the world seems to want a woman president.”

Some Trump supporters are less forthcoming. Last week in Houston, I was ejected from a Trump event hosted by an anti-immigration group that objected to my attempts to interview their supporters. Before I was asked to leave, a Trump supporter sidled up, making pointed comments to me about “snakes in the grass that bite people.”

Days later, at a county fair in southern Arizona, representatives of a local gun club refused point blank to speak to me. The only quote came from their T-shirts, which read: “Hillary For Prison.”

Yet many other voters were more thoughtful about their support for Mr. Trump. On Sunday, I went to see Dan Bell, a cattle rancher who lives on the Arizona border with Mexico, and whose land is frequently trespassed by migrants and drug smugglers. Mr. Bell was a classic rancher: a burly man with blue eyes, in a cowboy hat, quietly forthright. For him, Mr. Trump represents urgently needed change.

“Sure, there’s been things I wish he hadn’t said,” he told me as we bumped along the border in his Jeep, rain spitting against the windshield. “But this country needs a course correction. He’s been successful in business. We need to get the career politicians out.”

That is a common sentiment among Trump supporters: Their man fuses a potent anti-establishment vim with a certain brand of American authenticity. They may not agree with all of his more lurid attacks — although some clearly do — but they see in him the archetype of a successful if boastful businessman. To them he is someone whose words are to be taken with a pinch of salt.

Mr. Trump seems to have been stung by Monday’s debate, which was widely seen as a victory for Mrs. Clinton. Lashing out, he has blamed his performance on the debate moderator’s bias and a defective microphone, and threatened to turn Bill Clinton’s marital infidelity into an issue in the next debate, scheduled for Oct. 9 in St. Louis.

Convention political wisdom has it that Mr. Trump’s mercurial debate performance, while pleasing to his core supporters, did little to persuade other demographics, like women and college-educated whites, whose support he desperately needs. Mainstream Republicans are shunning him.

On Tuesday, The Arizona Republic, which has never endorsed a Democrat for president since it started publishing in 1890, formally declared for Mrs. Clinton. “The 2016 Republican candidate is not conservative and he is not qualified,” the paper said.

But the conventional wisdom has been wrong so many times in this election that only the foolish would dismiss Mr. Trump.

As people filed out of the debate party on Monday night in Phoenix, Phil Lovas, a member of the Arizona House of Representatives who heads the state Trump campaign, urged them to take home a Trump yard sign. “We’ve got 43 days until we elect President Trump,” he said. “We need your help.”

The man seated beside me, who had been silent for much of the debate, piped up. “President Trump,” he repeated, speaking the words in a low murmur, as if testing them out to see how they might sound.